E052: Totient ratio landscape

Preview figure for E052

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, totient, multiplicative See: Valid Tags.

Highlights

  • Plot \(\varphi(n)/n\) for a range of \(n\) and annotate record lows.

  • Relate deep dips to integers with many small prime factors (primorial-like structure).

    Goal

    Make the multiplicative structure of the totient function visible, especially the role of small primes.

    Background (quick refresher)

  • Euler’s totient function \varphi(n) refresher

  • Primorials

    Research question

    Which \(n\) minimize \(\varphi(n)/n\) up to a bound, and how do they relate to primorial products?

    Method

    • Compute \(\varphi(n)\) for \(n\le N\) (sieve / SPF-based).

  • Plot the ratio \(\varphi(n)/n\) and track record minima.

  • Optionally compare record minima to primorials and nearby multiples.

    How to run

    • make run EXP=e052

    • uv run python -m mathxlab.experiments.e052

    Outputs

    This experiment follows the standard output contract:

    • out/e052/figures/ — generated figures (PNG)

    • out/e052/report.md — short narrative report

    • out/e052/manifest.json — snapshot metadata for the gallery

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

  • n_max: 200000

  • min φ(n)/n in range: 0.191808 at n=30030

Figures:

  • fig_01_phi_over_n.png

  • fig_02_record_lows.png

params.json (snapshot)
{
  "n_max": 200000
}

References

See Apostol [1976], Tenenbaum [2015], Niven et al. [1991].